I know that if $K$ is a field, then $K[x]$ is an ED (Euclidean Domain), and that if $K[x]$ is a PID (Principal Ideal Domain) then $K$ is a field. Now, I can say that if $K[x]$ is an ED, then it is a PID, therefore $K$ is a field. Similarly, if $K$ is a field, then $K[x]$ it is an ED, therefore it is a PID. So basically I have that:
- (1) $K$ is a field $\Longleftrightarrow$ $K[x]$ is an ED
- (2) $K$ is a field $\Longleftrightarrow$ $K[x]$ is a PID
Putting these two things togheter I have that:
$K[x]$ is an ED $\Longleftrightarrow$ $K$ is a field $\Longleftrightarrow$ $K[x]$ is a PID
This brings to: $K[x]$ is an ED $\Longleftrightarrow$ $K[x]$ is a PID, which is obviously absurd.
I thought about possible mistakes in this argument, and the only explanation that I gave myself is that I used a sort of a circular argument, since to prove ($\Longleftarrow$) in (1) I used ($\Longleftarrow$) in (2), and to prove ($\Longrightarrow$) in (2) I used ($\Longrightarrow$) in (1). In conclusion, (1) and (2) are valid (it does make sense also because ED implies PID, so all the properties of a PID are also properties of an ED), while putting the implications together isn't.
Is that so? If yes, why? And if not, what is the problem?